The confidence of scientists at the end of the 19th century that we were close to an answer has been repeatedly rebuffed. While all sorts of forms of matter turn out to be theoretically possible, relatively few are stable.
Unnatural elements and isotopes
We have detected 118 chemical elements and have every reason to believe that elements with a larger number of protons are theoretically possible. But, no chemical element with an atomic number of 102 or greater has a single isotope with a half life of 1 day or greater.
No element with an atomic number greater than 82 (lead) is stable and not radioactive. Only four isotopes of three elements with a greater atomic number, Bismuth (atomic number 83, isotope 209), Thorium (90, isotope 232) and Uranium (92, isotopes 235 and 238), are found in nature. The most stable isotypes of each are radioactive, although this was just discovered in 2003 in the case of Bismuth. As Wikipedia explains:
232Th (thorium), 235U and 238U (uranium) are the only naturally occurring isotopes beyond bismuth that are relatively stable over the current lifespan of the universe. Bismuth was found to be hypothetically unstable in 2003, with an α-emission half-life of 1.9 × 1019 years for Bi-209. All other isotopes beyond bismuth are relatively or very unstable. So the main periodic table ends at bismuth, with an island at thorium and uranium. Between bismuth and thorium there is a sea trough of severe instability, which renders such elements as astatine, radon, and francium extremely short-lived relative to all but the heaviest elements found so far.
There are also two elements with atomic numbers lower than lead that do not have stable isotopes and are not found in nature. They are Technetium (atomic number 43) and Promethium (atomic number 61). Technetium was first synthesized and discovered in 1937 (some historians of science use a date a few years earlier which is disputed). Promethium and every chemical element with an atomic number of 93 or greater was first synthesized and discovered in 1940 or later. Science has only made these elements available to human experience for seventy years.
Thus, while it is possible to synthesize about 3,000 isotopes of at least 118 different chemical elements from protons, neutrons and electrons, less than 300 isotypes of 83 of these elements are found in nature, and there are only a handful of synthetic isotopes which are not found in nature but are as stable as the least stable isotopes found in nature.
Unstable Subatomic Particles
The quest to find out why big atoms are unstable, and why atomic nuclei hold together revealed yet more unstable types of matter. Subatomic particles other than the proton, neutron and electron were not discovered until the mid-20th century.
The standard model of particle physics, one of the iconic parts of modern quantum mechanics, has twelve particles it associates with ordinary matter called fermions. Half are leptons comprised of 3 kinds of neutrinos which come in two types each (particles and anti-particles) and 3 electron-like particles which come in two types each (particles and anti-particles). There are also 6 fermions called quarks, in six types each (red, green and blue in particle and anti-particle versions), which are the particles that combine to form particles such as protons and neutrons. Chirality almost doubles the number of types of particles in the standard model.
Quarks are never observed in isolation, they are always found in multiple quark structures called hadrons, almost always made of equal numbers of quarks and anti-quarks pairs of the same color (mesons), or one quark of each color (baryons). Unstable five quark structures may exist as well.
All quark structures containing heavier types of quarks than the up and down quarks (or their anti-particles) are unstable. All 140 or so of the two quark combinations (called mesons) are unstable with mean lifetimes on the order of a hundred millionth of a second or less. Likewise, all but four kinds of three quark combinations (called baryons) are extremely unstable with mean lifetimes of a tiny fraction of a second (on the order of one ten billionth of a second or less). The four stable baryons are protons and neutrons, which are composed solely of up and down quarks, and their anti-particles. Proton decay has not been observed and neutrons have a mean lifetime of just under 886 years.
The vast majority of stuff in the universe is composed of non-anti-matter up and down quarks connected in triplets that make protons and neutrons. Anti-matter has a nasty habit of swiftly transforming into pure energy upon contact with ordinary matter and is overwhelmingly outnumbered by ordinary matter in the universe.
Of the three electron-like particles, the two higher order versions are likewise unstable. See here and here.
The situation with neutrinos is complex, as they oscillate from one type to another. But, neutrinos, first observed in 1956, are hard to notice. According to Wikipedia: "Most neutrinos which pass through the Earth emanate from the sun and more than 50 trillion solar electron neutrinos pass through the human body every second."
On top of these are particles associated with fundamental forces, one for electromagnetism (photons), three for the nuclear weak force (two kinds of W and one kind of Z particle), and eight types of another for the nuclear strong force (gluons). Particles that account for interia mass (the Higgs boson) and for gravity (the graviton) have been hypothesized but not observed. All three particles associated with the nuclear weak force are very short-lived with a mean life of about 3 × 10^−25 second. Gluons are never observed in isolation, a property called confinement; they always bind multiple quark structures.
The longest lived massive subatomic particle for which a lifetime is meaningful, other than the proton, the neutron, and the electron, is the muon (i.e. a second order electron), which has a mean lifespan of 2.2 thousandths of a second. The only long lived observable force carrying particle is the photon.
Thus, nature provides an immensely complex array of hundreds of subatomic particles and combinations thereof, but leaves all but four so ephemeral that they are virtually impossible to observe without sophisticated scientific instrumentation.
The Uncertainty Principle
It is a basic tenant of quantum physics that it is theoretically impossible to ever know a full set of initial conditions for even a single quantum particle. There is dispute over whether the concept of a particle with an actual but unknown set of initial conditions is even a meaningful concept in quantum physics.
Meanwhile, at the other end of the universe where things are big, general and special relativity render what seems as if it should be simple, far more complex.
Phenomena that we currently call dark matter and dark energy that account for the vast majority of the matter-energy that general relativity implies must exist in the universe for it to behave as it appears to behave. But, we have never directly observed either, and have ruled out ordinary matter of any of the types described above as a possible source for more than a tiny fraction of dark matter and dark energy. Indeed, the only plausible candidate in quantum mechanics for dark energy (spontaneous formation and destruction of matter out of empty space sometimes called "zero point energy") produces back of napkin amounts of dark energy orders of magnitude out of whack with astronomical observations.
The more powerful a telescope you use, the farther back in time and the farther away from Earth you can see. We have seen objects many billions of years old, and many billions of light years from Earth. But, it is practically, and to some extent, even theoretically, it is impossible to see back to the moment of the presumed Big Bang, and while mainstream cosmology posits a finite universe, it is impossible to observe any boundaries of the universe directly.
We can look very far back in time and very far away, but ultimately are left guessing about where it all started, something irrevocably beyond our view, although we can get very, very close.
The world of the small (and the ancient, and the very large and the very distant), thus, has soft boundaries.
About 99.93% of stuff in the universe is made up of only ten chemical elements (just four, Hydrogen, Helium, Oxygen and Carbon make up more than 99%).
There are lots of kinds of atoms, but only about 10% are stable or only mildly radioactive.
There are lots of kinds of subatomic particles, but only 1% are stable and massive.
There are three forces understood at the quantum level, but only one operates outside the nuclear level.
The stuff we encounter in day to day life can be more or less fully understood with sophomore in college level mathematics. But, understanding the subatomic world, precisely as it really is, requires graduate student level mathematics.
There is an immense variety of possible kinds of matter, and yet just a handful of combinations overwhelmingly dominate the universe.
The deeper you dig, the more largely irrelevant forms of matter you discover. We live in a universe full of dead ends. We never seem to get to the bottom of everything, despite getting tantalizingly close.
To paraphrase the first scientists to observe the particle zoo, "who ordered this?"
UPDATED 12-5-2007 from an original post on 2-21-2007 at 6:29 P.M.